Synthesis and kinetic analysis of isothermal and non-isothermal decomposition of ammonium dinitramide prills

Article abstract of DOI:10.1007/s10973-007-8941-7

Ammonium dinitramide (ADN) prills were prepared by emulsion crystallization and characterized by optical microscopic, thermogravimetric (TG) and differential scanning calorimetric (DSC) techniques. The isothermal and non-isothermal decomposition kinetics of ADN prills were studied by TG. The differential isoconversional method of Friedman (FR) and integral isoconversional method of Vyazovkin were used to investigate the dependence of activation energy (E _a) with conversion (α) and the results were compared with literature data. The dependence of activation energy was also derived from isothermal data. A strong dependence of E _a with α is observed for the ADN prills. All the methods showed an initial increase in E _a up to α=0.2 and later decreases over the rest of conversion. The apparent E _a values of FR method are higher than that of Vyazovkin method up to α=0.45. The calculated mean E _a values by FR, Vyazovkin and standard isoconversional method for α between 0.05 and 0.95 were 211.0, 203.9 and 156.9 kJ mol^-1, respectively.

Full text of DOI:10.1007/s10973-007-8941-7

Journal of Thermal Analysis and Calorimetry, Vol. 94 (2008) 1, 263–270
SYNTHESIS AND KINETIC ANALYSIS OF ISOTHERMAL AND NON-ISO-
THERMAL DECOMPOSITION OF AMMONIUM DINITRAMIDE PRILLS
*
G. Santhosh and Ang How Ghee
Energetic Materials Research Centre, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639 798
Ammonium dinitramide (ADN) prills were prepared by emulsion crystallization and characterized by optical microscopic,
thermogravimetric (TG) and differential scanning calorimetric (DSC) techniques. The isothermal and non-isothermal decomposi-
tion kinetics of ADN prills were studied by TG. The differential isoconversional method of Friedman (FR) and integral
isoconversional method of Vyazovkin were used to investigate the dependence of activation energy (E
the results were compared with literature data. The dependence of activation energy was also derived from isothermal data. A
strong dependence of E up to a =~0.2 and
with a is observed for the ADN prills. All the methods showed an initial increase in E
later decreases over the rest of conversion. The apparent E values of FR method are higher than that of Vyazovkin method up to
a =~0.45. The calculated mean E values by FR, Vyazovkin and standard isoconversional method for a between 0.05 and 0.95
a
) with conversion (a ) and
a
a
a
a
–
1
were 211.0, 203.9 and 156.9 kJ mol , respectively.
Keywords: activation energy, ammonium dinitramide, emulsion crystallization, model-free kinetics, thermal decomposition
Introduction
Two different technologies were developed for
the formation of spherical ADN particles. The prilling
tower method as described by Highsmith et al.
whereby the molten ADN is sprayed through a nozzle
on the top of a huge glass column having hot and cold
zones and allow it to fall under gravity on a counter
current flow of cold inert gas [18, 19]. The molten
grains of ADN on contact with the cold gas solidify
and the same is collected at the bottom of the prilling
tower. In the second method as described by
Teipel et al. ADN is melted in hot paraffin oil and
stirred thoroughly until the molten ADN is uniformly
dispersed [20, 21]. The molten ADN grains are then
subsequently cooled to ambient temperature and the
formed prills are isolated from the oil and washed with
a solvent, the process is known as emulsion or melt
crystallization. The spherical ADN particles obtained
from the above methods are generally referred to as
ADN prills. The prills under our present study were
obtained from an emulsion crystallization process.
The ADN prills are reported to possess higher thermal
stability and are less sensitive than the crystalline
ADN resulting from the synthesis [22, 23] and the use
of bimodal mixtures of ADN prills of varying particle
Over the past few decades, there has been a rapidly
increasing interest and demand for the development
and use of new energetic materials for a variety of
space and military applications [1, 2]. A promising
oxidizer from the class of dinitramide i.e. ADN whose
structure shown in Fig. 1 is becoming more attractive as
potential replacements for ammonium perchlorate (AP)
and ammonium nitrate (AN) because of its predictable
burn rate, residue-less combustion, absence of halogen
and carbon atoms and high performance [3–5].
Studies on its synthesis, combustion and propellant
formulations have been published [6–9]. Theoretical
calculations indicate that solid propellant formulations
of ADN in various energetic binder systems will
outperform the conventional AP/HTPB formulations
in terms of specific impulse (Isp). To date, considerable
emphasis has been placed on studying the decomposition
kinetics of ADN and several works devoted to these
have been published [10–16]. The results from a
detailed analysis of the mechanism and decomposition
of ADN can be used to derive its combustion
characteristics. Despite being regarded as a high
performance oxidizer, ADN resulting from the synthesis
has undesirable crystal shape and poor morphology,
which prevents its use in propellant formulations and
is often necessary to tailor the crystal shape by proper
crystallization/prilling methods [17].
Fig. 1 Chemical structure of ADN
*
Author for correspondence: gsanthosh@ntu.edu.sg
1
388–6150/$20.00
Akadémiai Kiadó, Budapest, Hungary
Springer, Dordrecht, The Netherlands
©
2008 Akadémiai Kiadó, Budapest

SANTHOSH, GHEE
sizes in a propellant composition is expected to offer a
Methods
higher solid loading and a high theoretical maximum
density (TMD).
TG experiments were carried out using
a
Shimadzu TGA-50 thermogravimetric analyzer.
Non-isothermal runs were performed at constant heating
The solid-state reactions which involve a change in
extensive property such as mass (TG) and enthalpy (DSC)
have attracted wide interest as their kinetics can be
studied by thermoanalytical methods [24]. The kinetics
of such reactions is normally studied under isothermal
and/or non-isothermal conditions. The thermal analysis
of energetic materials is one of the prime areas which
is being explored in great detail. A traditional model
fitting approach using a single curve method has been
found to involve various curve-fitting techniques and
gives only a set of kinetic parameters (i.e. kinetic triplets
–
1
rates of 2, 3 and 6°C min . Isothermal runs were
performed at 125, 135, 145 and 150°C. Samples were
placed in aluminium pans and heated in a flowing
–1
nitrogen (50 mL min ) atmosphere. A constant sample
mass of 3.5–4 and 4.5–5 mg of samples were used for the
non-isothermal and isothermal experiments, respectively.
DTG curves were generated by smoothed numerical
derivatives of TG curves.
DSC measurements were performed using a
TA instruments DSC 2010 differential scanning
namely lnA, E
a
, f(a ) or g(a )). Reliable predictions on
–
1
calorimeter under a nitrogen flow of 50 mL min . A
constant sample mass of 2 to 2.5 mg is used for all the
experiments. Samples were taken in aluminium pans.
An Olympus BX-51 microscope was used for the
optical measurements of ADN prills.
the decomposition of a substance cannot be made
with this approach over the range of experimental
temperatures. The kinetic triplets from the model-fitting
technique from non-isothermal experiments are highly
uncertain and cannot be compared with the kinetic
triplets obtained from isothermal experiments.
The International Confederation of Thermal Analysis
and Calorimetry (ICTAC) project emphasizes the use
of multi-heating rate and isoconversional methods for
the kinetic computations [25]. The use of model-free
methods is a trustworthy way of obtaining reliable
and consistent kinetic information and increasingly
being used for energetic materials [26–28], polymers
The data analysis and kinetic calculations from
the TG measurements were done either by using
®
Microsoft EXCEL, Microcal ORIGIN or Mathworks
®
®
MATLAB softwares.
Results and discussion
Emulsion crystallization
[
29–31] and other materials [32]. The kinetic analyses
of the thermal decomposition of ADN by model
fitting and model-free isoconversional analysis were
reported in the literature [33–37]. However the thermal
decomposition kinetics of ADN prills has not been
studied so far. A thorough investigation on the ther-
mal decomposition of ADN prills will provide better
understanding of its physical and chemical character-
istics that affect the solid propellants during the
storage and use.
The process involves the melting of ADN in paraffin
oil maintained above the melting point of ADN
i.e. 92°C. The experimental setup is shown in Fig. 2.
Paraffin oil is taken in a flat glass vessel (GV) and placed
on a magnetic stirrer hotplate (HP). The temperature of
the paraffin oil is controlled by a thermocouple (TC)
immersed in the oil. The stirring of the oil and molten
ADN is achieved by placing a rectangular flat
In this paper, we describe a laboratory scale
process for producing ADN prills and the detailed
kinetic analyses of decomposition data of the prills.
The kinetic parameters and the apparent activation
energy on conversion evaluated by model-free methods
are presented.
Experimental
Materials
The ADN prills were prepared in the laboratory by
emulsion crystallization and their melting points were
determined by DSC. The samples were characterized
by spectral and thermal methods. Paraffin oil (J.T. Baker)
was used as received.
Fig. 2 Experimental setup for the emulsion crystallization of ADN
TC – thermocouple, GV – glass vessel, MB – metallic
bath, MP – magnetic pellet, HP – hotplate
264
J. Therm. Anal. Cal., 94, 2008

SYNTHESIS AND DECOMPOSITION OF AMMONIUM DINITRAMIDE PRILLS
magnetic pellet (MP). The whole assembly is placed
in a metallic bath (MB) which allows to quickly cool
the glass assembly with a coolant. Once the required
temperature of the paraffin oil i.e. 92–93°C is reached,
the recrystallized ADN is slowly introduced into the
paraffin oil under constant stirring.
ADN melts in paraffin oil and the stirring of the
oil aids in forming molten grains. After the complete
formation of the molten grains, the whole assembly is
slowly cooled to ambient temperature whereby the
molten ADN crystallizes to hard spherical grains. The
hard spherical grains were removed from the paraffin
oil by filtration and then washed few times with
methylene chloride and dried under vacuum.
Fig. 5 Optical microscopic images of ADN: a – prills and
b – recrystallized
of the melting point can be attributed either to the
thermal stability of ADN at its melting point or the
presence of trace amounts of the medium (paraffin
oil) during the emulsion crystallization process.
Comparison of the infrared (IR) spectra of ADN
Characterization of ADN prills
(prills and recrystallized) (Fig. 4) showed that both are
The ADN prills were then analysed by DSC. As seen
from Fig. 3, the melting point of ADN prills is
identical in shape and all the characteristic vibrations
were retained. Each characteristic vibrations of ADN
are marked with an asterisk in Fig. 4. The IR frequencies
of ADN can be found elsewhere [38].
9
1.89°C which is slightly lower than the recrystal-
lized ADN which is melting at 92.44°C. The lowering
The recrystallized ADN and the prills are subjected
to optical microscopic analysis. The optical microscopic
images of recrystallized and emulsion crystallized
ADN prills are shown in Fig. 5.
It can be seen that the recrystallized ADN have
irregular needle shaped crystals, while the emulsion
crystallized ADN prills are spherical. The particle size
of ADN prills as measured from optical microscopy is
in the range of 100 to 300 mm.
Thermal analysis
The thermal analyses of ADN prills were carried out
by TG to study the phenomenological aspects of
ADN. Figure 6 shows the TG-DTG curves of ADN
Fig. 3 DSC Melting traces of ADN: a – prills and
–1
prills at a heating rate of 2°C min ; similar curves
b – recrystallized
were obtained for all other heating rates.
Fig. 4 Overlay of IR spectrum of ADN: a – prills and
b – recrystallized
2
Fig. 6 TG and DTG curves of ADN prills in N atmosphere
J. Therm. Anal. Cal., 94, 2008
265

SANTHOSH, GHEE
Table 1 Characteristic temperatures of decomposition for
ADN prills from TG
The TG measurements showed that ADN prills
undergo single stage decomposition with a 100% mass
loss in the temperature range of 145 to 230°C, as it
gets converted to volatile decomposition products.
The well-defined DTG curve with a peak maximum at
T~181°C also indicates that it decomposes in a single
step. The literature reports indicate that the decomposition
of ADN is controlled by the experimental conditions
and different mechanisms were postulated by many
authors [11, 37, 39]. A generalized decomposition
mechanism shown in Scheme 1 suggests that the
decomposition of ADN produces various oxides of
nitrogen where the formation of ammonia and
Heating rate/
T
0
/°C
T
p
/°C
f
T /°C
a
max
–1
°C min
2
3
6
147.6
148.6
152.3
181.2
183.0
188.0
207.6
211.9
230.2
0.499
0.502
0.511
dinitramidic acid [HN(NO
below 100°C) followed by further decomposition of
the latter and AN formed by the recombination of
2 2
) ] dominate in the first step
(
NH
3
and HNO
3
at higher temperature (above 100°C).
NH N(NO ) ®
ü
ï
4
2 2
®
NH +HN(NO )
3
2
2
ï
low temperature
ý
HN(NO ) ®
2
2
ï
ï
®
HNO +N O
3
2
þ
Fig. 7 Variation of conversion (a ) with temperature at
various heating rates during the thermal
decomposition of ADN prills
NH +HNO ®
ü
ï
3
3
®
NH NO3
4
ï
tion of a with temperature (T) during the thermal
decomposition of ADN prills at various heating rates
is shown in Fig. 7.
NH NO ®
ï
4
3
ï
high temperature
ý
ï
®
N O+2H O
2
2
–
1
HN(NO ) ®
ï
It is seen from Fig. 7, that the run at 2°C min
2
2
ï
covers a temperature range of 150–205°C, where as
–1
the run at 6°C min covers about 150–230°C.
+
–
®
NO +HNNO ï
2
2
þ
Isothermal experiments were conducted at 125, 135,
Scheme 1 Thermal decomposition pathways of ADN
145 and 150°C on the ADN prills. The thermogravimetric
data showing the extent of conversion of ADN prills
during isothermal decomposition are shown in Fig. 8.
Taking into consideration the slow decomposition of
ADN above 125°C, isothermal experiments were
carefully performed and no conversion is observed
within the time gap when the samples were heated from
ambient to the experimental isothermal temperature.
As seen from Fig. 8, the extent of conversion is
low at 125°C and is high at higher temperatures. The
extent of conversion is quite rapid at 145 and 150°C,
which can be attributed to the exothermic decomposition
of ADN prills.
Apart from the decomposition fragments shown
in Scheme 1, many other products were also evolved
in varying ratios depending on the experimental
conditions. The thermal decomposition of ADN
reported by Brill et al. showed the presence of NH3,
N
2
O, NO
The values of initial temperature (T
temperature (T ) (from DTG), final temperature (T )
f
2
, H
2
O, NO, N
2
, HNO
3
and NH
4
NO
3
[39].
), peak
0
p
and the extent of reaction at maximum rate a max at
various heating rates are given in Table 1. The extent
of conversion (a ) is calculated from Eq. (1).
W –W
i
a
a =
(1)
W –W
Kinetic analysis
i
f
where, W
a
, W
i
and W
f
are the actual, initial and final
Dynamic and isothermal TG techniques are promising
tools to reveal the kinetics and mechanisms of the
physical and chemical processes occur during the
thermal decomposition of ADN. The kinetic analyses
have been carried out by model-free approaches. By
the application of model-free kinetics one can obtain
sample mass, respectively.
It is evident from Table 1 that the characteristic
decomposition temperatures of all the samples
showed a shift as the heating rate is increased and the
values of a max also showed a similar trend. The varia-
266
J. Therm. Anal. Cal., 94, 2008

SYNTHESIS AND KINETIC ANALYSIS OF ISOTHERMAL AND NONISOTHERMAL DECOMPOSITION OF
AMMONIUM DINITRAMIDE PRILLS
Fig. 8 Variation of conversion (a ) with time during the
isothermal decomposition of ADN prills. The
temperature (in°C) is indicated by each curve
a
Fig. 9 Apparent activation energy (E ) as a function of con-
version (a ) for the decomposition of ADN prills by
FR method
reliable and consistent kinetic information about the
overall process. The model-free kinetics assumes no
particular model in deriving the activation energy.
This approach follows all the conversion points from
multiple experiments performed at various heating
rates instead of a single experiment at a fixed heating
rate. The basic equation of non-isothermal kinetic
analysis (TG, DSC, etc.) is based on the rate equation
shown in Eq. (2).
heating rates, gives a straight line from whose slope
the activation energy can be calculated.
The dependence of activation energy as a function
of conversion by FR method is shown in Fig. 9.
It shows that the apparent E is not constant and it
a
–1
–1
increases from ~340 kJ mol (a =0.05) to ~393 kJ mol
–1
(
a =0.2) and decreases further to ~35 kJ mol (a =0.95).
The mean activation energy for conversion (0.05£a £0.2)
da
æ E ö
a
–1
b
= f (a )Aexpç –
÷
ø
(2)
is 370.2 kJ mol and for conversion (0.2<a £0.95) is
dT
RT
–1
76.9 kJ mol . The activation energy profile obtained
è
1
for the ADN prills indicates two distinct zones above
and below conversion 0.2 which could be attributed
to the two competing reaction pathways outlined in
Scheme 1.
where a is the degree of conversion, T is the temperature,
b is the linear heating rate, A is the pre-exponential
factor, Ea is the activation energy, f(a ) is the
differential function of conversion and R is the gas
constant. Starting from Eq. (2) various methods were
developed to evaluate the kinetic triplets. However
the use of isoconversional methods permit the
Vyazovkin’s method for kinetic analysis
Model-free kinetics method of Vyazovkin was used to
determine the apparent activation energy over conversion.
The general assumption used in Vyazovkin’s method
is that the pathway is independent of heating rate. For
a set of experiments carried out at different heating
rates, the activation energy Eaa can be determined at
any particular desired value of a by finding the value
of Eaa for which the function W is minimum as shown
in Eq. (4)
estimation of E without the prior knowledge of the
a
analytical form of the conversion function f(a ). The
non-isothermal data from multiple heating rates on
the decomposition of ADN prills were analyzed by
Friedman and Vyazovkin methods.
Friedman’s method
The method suggested by Friedman [40] shown in
Eq. (3) is obtained by simple rearrangement of
Eq. (2). It permits estimation of the activation energy
without the knowledge of the form of f(a ).
n
n
b I (E ,T )
j
aa
ai
W=
(4)
å å
i = l j¹ i
b
i
I (Eaa ,Taj
)
The indexes i and j in Eq. (4) denote different
heating rates, n is the total number of heating rates
and I is the temperature integral. The temperature in-
tegral I is given by Eq. (5)
æ da ö
lnç
E
a
÷
=ln[A f (a )] –
a
(3)
è dt ø
RT
a ,i
a , i
where subscript i denotes different heating rates. This
method requires numerical differentiation of the ex-
perimental a vs. T curves for each heating rate. For a
constant a , the plots of ln(da /dt) vs. 1/T, at several
T
ai
æ –E
ö
aa
I (E ,T )= p(x)= exp
ç
÷
dT
(5)
aa
ai
ò
è
RT
0
ø
J. Therm. Anal. Cal., 94, 2008
267

SANTHOSH, GHEE
The temperature integral in Eq. (5) can be evaluated
by various popular approximation methods and also
by direct numerical integration methods.
The temperature integral p(x), where x=E
can be approximated by a popular Senum and Yang
non-linear approximation [41]. In our studies fourth
four terms) rational approximation of Senum and
Yang as shown in Eq. (6) is used.
/RT
a
(
–
x
3
2
e
(x +18x +86x+96)
p(x)=
×
(6)
4
3
2
x (x +20x +120x +240x+120)
The minimization of Eq. (4) is done in
MATLAB using the medium scale quasi-Newton
method with a mixed quadratic and cubic line search
procedure. The in-built function ‘fminunc’ is applied
for the optimization.
The plot of dependence of activation energy with
conversion is shown in Fig. 10.
Fig. 11 Dependence of E
a
on a for FR (ADN prills: l – b=2,
–1
3
and 6°C min ); Vyazovkin’s (ADN prills: r – b=2,
3 and 6°C min ) and literature data (ADN:
It can be seen from the Fig. 10 that the E
–
a
–1
1
increases to a maximum of 308 kJ mol around 28%
conversion, thereafter it decreases over the rest of the
conversion. The trend and shape of the apparent
activation energy computed by Vyazovkin method is
similar to that of FR method. The mean activation
–
1
o – b=1.5, 4 and 5.5°C min )
at conversion a =~0.2 (dotted lines). At conversions
above 0.2, the activation energy decreases very fast.
For conversions upto 0.45, the activation energies
obtained by FR method are higher than that of Vyazovkin
–
1
energy for conversion (0.05£a £0.2) is 268.1 kJ mol
–
1
and for conversion (0.2<a £0.95) is 180.9 kJ mol .
method and at conversions above 0.65, the E values
a
by Vyazovkin method are higher. The susceptibility
of the Friedman’s method to errors arising from
experimental noise and also the assumption of
constant E over the conversion in Vyazovkin’s method
a
can be contributed to the observed difference in the Ea
profiles. A comparison of our results on the dependence
of activation energy for ADN prills with that of the
results by Vyazovkin et al. shows that the E values
a
vary significantly with conversion and the E values
a
for the prills are much higher than that of the values
obtained by Vyazovkin et al. for ADN. The E profile
a
for the literature data at conversions above 0.2
deviates from that of ADN prills. The higher
activation energy for the ADN prills clearly shows
that it possesses higher thermal stability.
Fig. 10 Dependence of activation energy on conversion for
the non-isothermal decomposition of ADN prills using
Vyazovkin method
Isoconversional method for isothermal data
The dependence of activation energy (E ) with
a
Comparison of results from FR and Vyazovkin method
conversion (a ) is derived using standard
isoconversional method [34] for the thermal
decomposition of ADN prills under isothermal
conditions. This is derived from the isothermal rate
law as shown in Eq. (7)
The calculated values of the apparent activation energies
by FR and Vyazovkin methods for ADN prills are
presented together with the literature data for ADN in
Fig. 11. The data for ADN was taken from [33]. The
dependence of E
a
with a is consistent with the
–Ea
RT
reported trend observed for neat crystalline ADN by
Vyazovkin et al. [33, 37].
g(a )= Ae
t
(7)
where t is the time and g(a ) is the reaction model.
Taking logarithm of Eq. (7) one can get Eq. (8).
It can be seen from Fig. 11 that the isoconversional
methods have showed a maximum activation energy
268
J. Therm. Anal. Cal., 94, 2008

SYNTHESIS AND KINETIC ANALYSIS OF ISOTHERMAL AND NONISOTHERMAL DECOMPOSITION OF
AMMONIUM DINITRAMIDE PRILLS
Ea
ADN prills. The dependence of E with a by all the
a
lng(a )=lnA–
+ lnt
(8)
methods is similar in shape and the apparent E values
RT
a
for conversion (0.2<a £0.95) are in the order of
Vyazovkin >FR>standard isoconversional method. In
the initial stages of decomposition (0.05£a £0.2), the
Ea value increases and in the latter stages (0.2<a £0.95)
it decreases. A comparison of the results obtained
indicates that the thermal stability of ADN prills was
much higher than that of neat crystalline ADN.
For each a , rearrangement of Eq. (8) gives Eq. (9).
é A ù
(9)
E
aa
–
lnt =ln
–
a
ê
ú
g(a )
ë
RT
û
where t is the time at conversion a . Under isothermal
a
conditions for each a , Eaa is calculated from the slope
of the plot of –lnt
Equation (9) has been applied to the isothermal data
for the decomposition of ADN prills (data from
Fig. 8) which permits the determination of E as a
a
vs. 1/T regardless of the model.
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a
1
2
3
J. C. Bottaro, Chem. Ind., (1996) 249.
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ADN prills is shown in Fig. 12.
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1
1
0 M. J. Rossi, J. C. Bottaro and D. F. McMillen, Int.
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1 J. C. Oxley, J. L. Smith, W. Zheng, R. Rogers and
M. D. Coburn, J. Phys. Chem. A, 101 (1997) 5646.
It is seen that the E
values bear a strong relation-
a
ship to the degree of conversion. The activation energy
–
1
12 A. S. Tompa, R. F. Boswell, P. Skahan and C. Gotzmer,
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rises from about 100 kJ mol at low conversion to
–
about 165 kJ mol at a =0.27 and it subsequently de-
creases to 145 kJ mol near the completion of the re-
1
1
1
–
1
1
action. The mean activation energy for conversion
–
1
(
0.05£a £0.27) is 151.9 kJ mol–1 and for conversion
0.27<a £0.95) is 156.3 kJ mol which is lower than
15 G. Santhosh, S. Venkatachalam, A. U. Francis,
rd
K. Krishnan, K. B. Catherine and K. N. Ninan, 33 Int.
Annu. Conf. ICT, 2002, p. 64-1.
(
that of the values obtained by FR and Vyazovkin
method. In the present study a comparison of the de-
pendencies of E
on a by non-isothermal and isother-
mal methods should not be made because the former
covers a wide range of temperature (120 to 230°C)
while the latter covers a narrow range of temperature
1
1
1
6 G. Santhosh, S. Venkatachalam, K. Krishnan,
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ADN prills were prepared by emulsion crystallization
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2
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a
with a at various conversions for
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